Area between a Line and the Graph of a Function

Find the slope of the line that divides the area under the sine curve between zero and into two equal parts. Or, find the slope of the line that divides the area under the circle between zero and two into two equal parts.

The area to be bisected in the case of the sine curve is , which is 1. The line intersects the sine curve at the point , so . Given , Mathematica solves this nonalgebraic equation for to find the point of intersection. The shaded area consists of two parts: the triangular part with area and the area under the sine curve from to , .

The area to be bisected in the case of the circle is , half of which is about . The shaded area is equal to the area of the triangle with vertices the origin, the point of intersection, and the center of the circle plus the area of the sector with an angle equal to twice the angle of elevation of the line. The shaded area is equal to .

Checking the answer key gives the coordinates of the intersection points and the area of the shaded region.